crit_logEI: Logarithm of Expected Improvement criterion

Description

Computes log of EI for minimization, with improved stability with respect to EI

Usage

crit_logEI(x, model, cst = NULL, preds = NULL)

Arguments

x: matrix of new designs, one point per row (size n x d)
model: homGP or hetGP model, or their TP equivalents, including inverse matrices. For TP models, the computation is using the one from regular EI.
cst: optional plugin value used in the EI, see details
preds: optional predictions at x to avoid recomputing if already done

Details

cst is classically the observed minimum in the deterministic case. In the noisy case, the min of the predictive mean works fine.

References

Ament, S., Daulton, S., Eriksson, D., Balandat, M., & Bakshy, E. (2024). Unexpected improvements to expected improvement for Bayesian optimization. Advances in Neural Information Processing Systems, 36.

Examples

Run this code

## Optimization example
set.seed(42)


## Noise field via standard deviation
noiseFun <- function(x, coef = 1.1, scale = 1){
if(is.null(nrow(x)))
 x <- matrix(x, nrow = 1)
   return(scale*(coef + cos(x * 2 * pi)))
}

## Test function defined in [0,1]
ftest <- function(x){
if(is.null(nrow(x)))
x <- matrix(x, ncol = 1)
return(f1d(x) + rnorm(nrow(x), mean = 0, sd = noiseFun(x)))
}

n_init <- 10 # number of unique designs
N_init <- 100 # total number of points
X <- seq(0, 1, length.out = n_init)
X <- matrix(X[sample(1:n_init, N_init, replace = TRUE)], ncol = 1)
Z <- ftest(X)

## Predictive grid
ngrid <- 51
xgrid <- seq(0,1, length.out = ngrid)
Xgrid <- matrix(xgrid, ncol = 1)

model <- mleHetGP(X = X, Z = Z, lower = 0.001, upper = 1)

logEIgrid <- crit_logEI(Xgrid, model)
preds <- predict(x = Xgrid, model)

par(mar = c(3,3,2,3)+0.1)
plot(xgrid, f1d(xgrid), type = 'l', lwd = 1, col = "blue", lty = 3,
xlab = '', ylab = '', ylim = c(-8,16))
points(X, Z)
lines(Xgrid, preds$mean, col = 'red', lwd = 2)
lines(Xgrid, qnorm(0.05, preds$mean, sqrt(preds$sd2)), col = 2, lty = 2)
lines(Xgrid, qnorm(0.95, preds$mean, sqrt(preds$sd2)), col = 2, lty = 2)
lines(Xgrid, qnorm(0.05, preds$mean, sqrt(preds$sd2 + preds$nugs)), col = 3, lty = 2)
lines(Xgrid, qnorm(0.95, preds$mean, sqrt(preds$sd2 + preds$nugs)), col = 3, lty = 2)
par(new = TRUE)
plot(NA, NA, xlim = c(0, 1), ylim = range(logEIgrid), axes = FALSE, ylab = "", xlab = "")
lines(xgrid, logEIgrid, lwd = 2, col = 'cyan')
axis(side = 4)
mtext(side = 4, line = 2, expression(logEI(x)), cex = 0.8)
mtext(side = 2, line = 2, expression(f(x)), cex = 0.8)